Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

Abstract

A Banach space operator $T\in B({\cal X})$ is polaroid if points $\lambda\in\iso\sigma\sigma(T)$ are poles of the resolvent of $T$. Let $\sigma_a(T)$, $\sigma_w(T)$, $\sigma_{aw}(T)$, $\sigma_{SF_+}(T)$ and $\sigma_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $C\in B({\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $\pi_0(M_C)=\{\lambda\in\iso\sigma(M_C) 0<\dim(M_C-\lambda)^{-1}(0)<\infty\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$ and $B$ has SVEP at points $\mu\in\sigma_w(M_0)\setminus\sigma_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$ and $B^*$ has SVEP at points $\mu\in\sigma_w(M_0)\setminus\sigma_{SF_-}(B)$, then $\sigma(M_C)\setminus\sigma_w(M_C)=\pi_0(M_C)$. Here the hypothesis that $\lambda\in\pi_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $\lambda\in\pi_0(A)$ are poles of the resolvent of $A$. For an operator $T\in B(\X)$, let $\pi_0^a(T)=\{\lambda:\lambda\in\iso\sigma_a(T), 0<\dim(T-\lambda)^{-1}(0)<\infty\}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $\pi_0^a(\M)$ and $B$ is polaroid on $\pi_0^a(B)$, then $\sigma_a(\M)\setminus\sigma_{aw}(\M)=\pi_0^a(\M)$.